Related papers: Power majorization between the roots of two polyno…
In parameter slices of quadratic rational functions, we identify arcs represented by matings of quadratic polynomials. These arcs are on the boundaries of hyperbolic components.
We propose a geometric explanation for the observation that generic quadratic polynomials over split quaternions may have up to six different factorizations while generic polynomials over Hamiltonian quaternions only have two. Split…
We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing…
We investigate some Galois groups of linearized polynomials over fields such as $\mathbb{F}_q(t)$. The space of roots of such a polynomial is a module for its Galois group. We present a realization of the symmetric powers of this module, as…
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…
In this paper methods for simultaneous finding all roots of generalized polynomials are developed. These methods are related to the case when the roots are multiple. They possess cubic rate of convergence and they are as labour-consuming as…
A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where…
We pose the question of what is the best generalization of the factorial and the binomial coefficient. We give several examples, derive their combinatorial properties, and demonstrate their interrelationships. On cherche ici \`a…
The hyperdeterminant of a polynomial (interpreted as a symmetric tensor) factors into several irreducible factors with multiplicities. Using geometric techniques these factors are identified along with their degrees and their…
Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a…
We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from $\mathbb{F}_q$-linear formal power series. Since the logarithmic derivatives connect…
We characterize all possible relative positions between a hyperboloid of one sheet and a sphere through the roots of a characteristic polynomial associated to these quadrics. The classification is also suitable for a hyperboloid and a…
It is shown that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has positive coefficients when $m = 6$ and $A$ and $B$ are any two 3-by-3 complex Hermitian positive definite matrices. This case is the first that is not covered by prior,…
We study a variant of the majorization relation. In particular we consider inequalities involving some Schur-concave symmetric polynomials related to the multinomial expansion. We also discuss how these topics were motivated by conjectures…
We use the properties of Hermite and Kamp\'e de F\'eriet polynomials to get closed forms for the repeated derivatives of functions whose argument is a quadratic or higher-order polynomial. The results we obtain are extended to product of…
The polynomials that arise as coefficients when a power series is raised to the power $x$ include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such…
The non-empty finite subsets of a multiplicatively written monoid form a monoid under setwise multiplication. The same holds for finite subsets containing the identity element. Partly due to their unusual arithmetic properties, these…
It is known that the weight (that is, the number of nonzero coefficients) of a univariate polynomial over a field of characteristic zero is larger than the multiplicity of any of its nonzero roots. We extend this result to an appropriate…
We present a proof of the Casas-Alvero conjecture, stating that if a complex polynomial has a root in common with each of its derivatives it must be a multiple of the power of some monomial.
The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…